Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Sebastian Casalaina-Martin

Second Advisor

Jonathan Wise

Third Advisor

Su-ion Ih

Fourth Advisor

Jeff Achter

Fifth Advisor

Samouil Molcho

Abstract

In this thesis we describe intermediate Jacobians of threefolds obtained from singular cubic threefolds. By this we mean two things. First, we describe the intermediate Jacobian of a desingularization of a cubic threefold with isolated singularities. Second, we describe limits of intermediate Jacobians of smooth cubic threefolds, as the family of cubic threefolds acquires isolated singularities. In regards to the first question, generalizing a result of Clemens--Griffiths we show specifically that the intermediate Jacobian of a distinguished desingularization of a cubic threefold with a single singularity of type $A_3$ is the Jacobian of the normalization of an associated complete intersection curve in $\mathbb P^3$, the so called $(2,3)$-curve. In regards to degenerations, we describe how the limit intermediate Jacobian, under certain conditions, can be described as a semi-abelian variety as the extension of a torus by the finite quotient of the product of Jacobians of curves, where one of the curves is the normalization of the $(2,3)$-curve associated to the cubic threefold and a choice of singularity, and the other curves are so-called tails arising from stable reduction of plane curve singularities.

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